The optimal packing of spheres in higher dimensions is an open problem with important implications for computing and error-correcting code in communications. It might seem that the best arrangement should involve some sort of regular lattice – think of stacking oranges in the market – and indeed in three dimensions the face-centre cubic lattice provides the densest packing arrangement. This was conjectured by Johannes Kepler some 400 years ago but proved only recently.

When it comes to higher dimensions the case is less clear, and no-one has yet improved on the lower bound on maximal packing density found 100 years ago by Hermann Minkowski for a d-dimensional space. Now, however, Salvatore Torquato and Frank Stillinger at Princeton University have found that sometimes it may be better to vary the local density of packing – that is, to pack in a disordered way. They argue that for a high number of dimensions lattice packings are unlikely to be the densest, with disordered packing becoming more important as the number of dimensions increases.

Further reading

S Torquato and F H Stillinger 2006 Experimental Mathematics 15 307.